The sand timer in the above figure has a due time of \( T. \) The radius of the hole is \( r, \) the initial height of the sand is \( H, \) and the density of the sand is \( \rho. \)
Using the Buckingham \(\pi\) theorem, determine \( T \) in terms of the properties \( r, H, \rho, \) and the gravitational constant \( G .\)

As shown in the above figure, a liquid with density \( \rho \) flows through a pipe with diameter \( d. \) In a section of the pipe with length \( L, \) the liquid flows with a speed of \( v_1 \) at the center and \( v_2 \) at the edge of the flow. The pressures are \( P_1 \) at the center and \( P_2 \) at the edge. Using the Buckingham \(\pi\) theorem, express the viscosity \( \mu \) of the fluid in terms of the liquid properties \( d, L, \rho, \Delta v = v_1 - v_2 \) and \( \Delta P = P_1- P_2 .\)

The drag force \( F \) depends on four quantities:
two parameters of the cone which are the speed of the cone \( v \) and the size of cone \( r,\) and two parameters of the air which are the density of the air \( \rho \) and the viscosity of the air \( \mu. \) Find the independent dimensionless groups that can be produced with \( F, v, r, \rho \) and \( \mu . \)